(Choosing numbers) Players 1 and 2 each choose a positive integer up to K. If the players choose…

(Choosing numbers) Players 1 and 2 each choose a positive integer up to K. If the players choose the same number then player 2 pays $1 to player 1; otherwise no payment is made. Each player’s preferences are represented by her expected monetary payoff.

a. Show that the game has a mixed strategy Nash equilibrium in which each player chooses each positive integer up to K with probability 1/K.

b. (More difficult.) Show that the game has no other mixed strategy Nash equilibria. (Deduce from the fact that player 1 assigns positive probability to some action k that player 2 must do so; then look at the implied restriction on player 1’s equilibrium strategy.)